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Some More Aspects of Quantum Information Theory

Harish Parthasarathy

Published Nov 1, 2022• 0 citations

Abstract

Let ρ, σ be two states and let spectral be https://www.w3.org/1998/Math/MathML"> ρ = ∑ a | e a > p ( a ) < e a | , σ = ∑ a | f a > q ( a ) < f a | https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353553/1603a02c-0268-4c9d-8b97-09135b05559e/content/math10_129_1_B.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> so that < e a | e b >= δ ( a, b ), < f a | f b >= δ ( a, b ). The relative entropy between these two states is https://www.w3.org/1998/Math/MathML"> D ( ρ | σ ) = T r ( ρ . ( log ( ρ ) − log ( σ ) ) ) = ∑ a p ( a ) log ( p ( a ) ) − ∑ a , b p ( a ) log ( q b ) | < e a | f b > | 2 = ∑ a , b ( p ( a ) | < e a | f b > | 2 ( log ( p ( a ) | < e a | f b > | 2 ) − log ( q b | < e a | f b > | 2 ) ) = ∑ a , b ( P ( a , b ) log ( P ( a , b ) ) − log ( Q ( a , b ) ) ) = D ( P | Q ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353553/1603a02c-0268-4c9d-8b97-09135b05559e/content/math10_129_2_B.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/>

DOI: 10.1201/9781003353553-9

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In the previous chapter, we did not examine the change of a state with time https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . It has been assumed that we have been examining a state at a given moment https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_2.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . In the general case, the state of a system changes with time https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_3.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . For example, say the motion of an electron in a potential field, its wave function depends on time https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_4.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> : https://www.w3.org/1998/Math/MathML"> Ψ = Ψ q , t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_5.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . Such a dependence has to be described by a differential equation in which we may have https://www.w3.org/1998/Math/MathML"> ∂ Ψ ∂ t , ∂ 2 Ψ ∂ t 2 , ⋯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_6.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> .